### Resumo:

In recent years, a family of probability distributions based on Nonextensive Statistical
Mechanics, known as q-distributions, has experienced a surge in terms of applications to
several fields of science and engineering. In this work the _-Exponential distribution will be
studied in detail. One of the features of this distribution is the capability of modeling data that
have a power law behavior, since it has a heavy-tailed probability density function (PDF) for
particular values of its parameters. This feature allows us to consider this distribution as a
candidate to model data sets with extremely large values (e.g. cycles to failure). Once the
analytical expressions for the maximum likelihood estimates (MLE) of _-Exponential are
very difficult to be obtained, in this work, we will obtain the MLE for the parameters of the _-
Exponential using two different optimization methods: particle swarm optimization (PSO)
and Nelder-Mead (NM), which are also coupled with parametric and non-parametric
bootstrap methods in order to obtain confidence intervals for these parameters; asymptotic
intervals are also derived. Besides, we will make inference about a useful performance metric
in system reliability, the called index __(_, where the stress _ and strength are
independent q-Exponential random variables with different parameters. In fact, when dealing
with practical problems of stress-strength reliability, one can work with fatigue life data and
make use of the well-known relation between stress and cycles until failure. For some
materials, this kind of data can involve extremely large values and the capability of the q-
Exponential distribution to model data with extremely large values makes this distribution a
good candidate to adjust stress-strength models. In terms of system reliability, the index _ is
considered a topic of great interest, so we will develop the maximum likelihood estimator
(MLE) for the index _ and show that this estimator is obtained by a function that depends on
the parameters of the distributions for and _. The behavior of the MLE for the index _ is
assessed by means of simulated experiments. Moreover, confidence intervals are developed
based on parametric and non-parametric bootstrap. As an example of application, we consider
two experimental data sets taken from literature: the first is related to the analysis of high
cycle fatigue properties of ductile cast iron for wind turbine components, and the second one
evaluates the specimen size effects on gigacycle fatigue properties of high-strength steel.